On a class of nonlinear Schrödinger equations. I. The Cauchy problem, general case. (English) Zbl 0396.35028

Ginibre, J.; Velo, G.

J. Funct. Anal. 32, 1-32 (1979).

MSC:

35J10	Schrödinger operator, Schrödinger equation
35J60	Nonlinear elliptic equations
35D05	Existence of generalized solutions of PDE (MSC2000)
35A05	General existence and uniqueness theorems (PDE) (MSC2000)

Keywords:

Cauchy Problem; Nonlinear Schrödinger Equations; Global Solutions; Initial Value Problem; Sobolev Space

References:

[1]	Dunford, N.; Schwartz, J., ()
[2]	de Gennes, P.G., Superconductivity of metals and alloys, (1966), Benjamin New York, Chap. 6 · Zbl 0138.22801
[3]	Hille, E.; Phillips, R.S., Functional analysis and semi-groups, (1957), American Mathematical Society Providence R.I
[4]	Scott, A.C.; Chu, F.Y.F.; McLaughlin, D.W., The soliton: A new concept in applied science, (), 1143-1483
[5]	Strauss, W.A., Nonlinear scattering theory, (), 53-78 · Zbl 0297.35062
[6]	Volevic, L.R.; Paneyakh, B.P., Certain spaces of generalized functions and embedding theorems, Russian math. surveys, 20, 1-73, (1965)
[7]	Zakharov, V.E.; Shabat, A.B., Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media, Sov. physics JETP, 34, 62-69, (1972)
[8]	Baillon, J.B.; Cazenave, T.; Figueira, M., C. R. acad. sci. Paris, 284, 869-872, (1977)
[9]	Glassey, R.T., On the blowing up of solutions to the Cauchy problem for nonlinear Schrödinger equations, J. math. phys., 18, 1794-1797, (1977) · Zbl 0372.35009
[10]	{\scJ. E. Lin and W. Strauss}, Decay and scattering of solutions of a non-linear Schrödinger equation, to appear.
[11]	Lions, J.L.; Magenes, E., ()